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放缩不等式推导

污浊的双黑 2023-06-01 16:00:02
简介放缩不等式推导

放缩不等式推导

1 )   a x > x + 1 ( 1 < a ≤ e , x < 0 ; a ≥ e , x > 0 ) ; 1) a^x>x+1left(1<aleq e,x<0;ageq e,x>0 ight); 1) ax>x+1(1<ae,x<0;ae,x>0);
p r o o f : proof: proof:
f 01 ( x ) = a x − ( x + 1 ) ⇒ f 01 ′ ( x ) = a x ln ⁡ a − 1 f_{01}left(x ight)=a^{x}-left(x+1 ight)Rightarrow f_{01}^{'}left(x ight) = a ^{x} ln a-1 f01(x)=ax(x+1)f01(x)=axlna1
1 < a ≤ e , x < 0 ⇒ 0 < a x < 1 , 0 < ln ⁡ a ≤ 1 ⇒ f 01 ( x ) > f 01 ( 0 ) = 1 − 1 = 0 ⇒ a x > x + 1 ( 1 < a ≤ e , x < 0 ) ; 1<aleq e,x<0\Rightarrow0<a^{x}<1,0<ln aleq1\Rightarrow f_{01}left(x ight)>f_{01}left(0 ight)=1-1=0\Rightarrow a^{x}>x+1left(1<aleq e,x<0 ight); 1<ae,x<00<ax<1,0<lna1f01(x)>f01(0)=11=0ax>x+1(1<ae,x<0);

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